1. Field of the Invention
The present invention relates to a method, an apparatus, and a program for restoring phase information used for constructing an image based on image information obtained by radiation imaging etc. Note that, in this application, the term “radiation” indicates radiation in a broad sense including a corpuscular beam such as an electron beam, and an electromagnetic wave, in addition to common radiation such as an X-ray, an α-ray, a β-ray, a γ-ray, and ultraviolet rays.
2. Description of a Related Art
Conventionally, an imaging method using an X-ray etc. is utilized in various fields, and particularly, in the medical field, the method is one of the most important means for diagnosis. Since the first X-ray photograph was implemented, the X-ray photography method has been repeatedly improved, and a method using a combination of a fluorescent screen and an X-ray film is in the mainstream at present. On the other hand, in recent years, various digitized devices applying X-ray CT, ultrasonic waves, MRI, etc. are in practical use, and construction of a diagnostic information processing system etc. in hospitals is being promoted. Many studies are also made for digitizing the imaging system with regard to X-ray images. Digitizing the imaging system enables to store a large amount of data for a long period without degradation in image, and helps to make progress toward the medical diagnostic information processing system.
Now, a radiation image obtained as described above is generated by converting intensity of the radiation transmitted through an object into brightness of the image. For example, when performing imaging on a region including a bone part, the radiation transmitted through the bone part is largely attenuated, and the radiation transmitted through a region other than the bone part, i.e., a soft part is slightly attenuated. In this case, since the difference in intensity between the radiation transmitted through different tissues is large, a high-contrast radiation image can be obtained.
On the other hand, for example, when imaging a region of the soft part such as a breast, since radiation is easily transmitted through the soft part as a whole, the difference between tissues in the soft part is difficult to appear as the difference in intensity of the transmitted radiation. Therefore, only a low-contrast image can be obtained with respect to the soft part. Thus, the radiation imaging method is not suitable as a method of visualizing the slight difference between tissues in the soft part.
Here, as information included in the radiation transmitted through the object, there is phase information in addition to intensity information. Recently, a phase-contrast method of generating an image using this phase information is under study. The phase-contrast method is a phase information restoration technology to convert phase difference produced by an X-ray etc. transmitted through the object into brightness of an image.
The phase-contrast method includes techniques for obtaining phase difference based on interference X-rays produced by using an interferometer or a zone plate, and for obtaining phase difference based on diffracted X-rays. Of these techniques, the diffraction technique for obtaining phase difference based on diffracted X-rays is to obtain phase difference according to the following principle. An X-ray, for example, propagates within a material since a wave progresses similarly to light. The propagation velocity thereof varies in accordance with the refractive index the material has. Therefore, when irradiating an object with an X-ray that has a uniform phase, the way the X-ray propagates varies in accordance with the difference between tissues in the object. Thereby, the wavefront of the X-ray transmitted through the object is distorted and, as a result, diffraction fringes are produced on the X-ray image obtained based on the transmitted X-ray. The pattern of the diffraction fringes differs in accordance with the distance between the screen on which the X-ray image is formed and the object, and the wavelength of the X-ray. Thus, analyzing two or more X-ray images having different patterns of diffraction fringes, phase differences of the X-ray produced on the respective positions on the screen can be obtained. Converting the phase differences into brightness, an X-ray image that clearly shows the difference between tissues in the object can be obtained.
Particularly, in the radiation after transmitted through a soft part of the object, since the difference in phase is larger than the difference in intensity in accordance with the difference between tissues through which the radiation has been transmitted, the subtle difference between tissues can be visualized by using the phase-contrast method.
In order to use the above phase-contrast method, imaging conditions in radiation imaging or techniques for restoring phase from patterns of diffraction fringes are under study.
For example, B. E. Allman et al. “Noninterferometric quantitative phase imaging with soft x rays”, J. Optical Society of America A, Vol. 17, No. 10 (October 2000), pp. 1732–1743, discloses that an X-ray image is constructed by performing phase restoration based on image information obtained by imaging with soft X-rays.
In this reference, the basic expression of phase restoration, TIE (transport of intensity equation) is used.
                              κ          ⁢                                    ∂                              I                ⁡                                  (                                      r                    ⇀                                    )                                                                    ∂              z                                      =                              -            ∇                    ⊥                      ·                          {                                                                    I                    ⁡                                          (                                              r                        ⇀                                            )                                                        ∇                                ⊥                                  ϕ                  ⁡                                      (                                          r                      ⇀                                        )                                                              }                                                          (        1        )            Where
      ∇    ⊥    =      (                  ∂                  ∂          x                    ,                          ⁢              ∂                  ∂          y                      )  In addition, κ denotes wave number.
Here, a principle of the phase restoration will be described by referring to FIG. 11. As shown in FIG. 11, an X-ray having a wavelength λ is output from the left side of the drawing, transmitted through an object plane 101 and enters a screen 102 at a distance of z from the object plane 101. Assuming that the intensity and the phase of the X-ray at a position (x,y) on the screen 102 are I(x,y) and φ(x,y), respectively, a relationship between intensity I(x,y) and phase φ(x,y) is expressed by the following expression. Here, the intensity I is square of amplitude of wave.
                                                        2              ⁢              π                        λ                    ⁢                                    ∂                              I                ⁡                                  (                                      x                    ,                                                                                  ⁢                    y                                    )                                                                    ∂              z                                      =                              -            ∇                    ·                      {                                          I                ⁡                                  (                                      x                    ,                                                                                  ⁢                    y                                    )                                            ⁢                              ∇                                  ϕ                  ⁡                                      (                                          x                      ,                                                                                          ⁢                      y                                        )                                                                        }                                              (        2        )            Substituting κ=2π/λ into expression (2) and rewriting (x,y) components into vector r, TIE expressed by expression (1) is derived.
However, since the above TIE is difficult to be solved, TIE has been used mainly by performing approximation thereon. For example, T. E. Gureyev et al. “Hard X-ray quantitative non-interferometric phase-contrast imaging”, SPIE Vol. 3659 (1999), pp. 356–364, discloses that an X-ray image is constructed by performing phase restoration based on image information obtained by imaging with hard X-rays.
In this reference, TIE expressed by expression (1) is approximated as follows. First, expression (1) is developed. Note that, in the following expressions, the vector r in the above reference is rewritten into (x,y) components.
                                          -            κ                    ⁢                                    ∂                              I                ⁡                                  (                                      x                    ,                                                                                  ⁢                    y                                    )                                                                    ∂              z                                      =                              (                                                            ∂                                                                                                          ∂                  x                                            ,                                                          ⁢                              ∂                                  ∂                  y                                                      )                    ·                      (                                                            I                  ⁡                                      (                                          x                      ,                                                                                          ⁢                      y                                        )                                                  ⁢                                                      ∂                                          ϕ                      ⁡                                              (                                                  x                          ,                                                                                                          ⁢                          y                                                )                                                                                                  ∂                    x                                                  ,                                                                  ⁢                                  I                  ⁡                                      (                                          x                      ,                                                                                          ⁢                      y                                        )                                                  ⁢                                                      ∂                                          ϕ                      ⁡                                              (                                                  x                          ,                                                                                                          ⁢                          y                                                )                                                                                                  ∂                    y                                                              =                                                                                          ∂                                              ∂                        x                                                              ⁢                                          (                                                                        I                          ⁡                                                      (                                                          x                              ,                                                                                                                          ⁢                              y                                                        )                                                                          ⁢                                                                              ∂                                                          ϕ                              ⁡                                                              (                                                                  x                                  ,                                                                                                                                          ⁢                                  y                                                                )                                                                                                                                          ∂                            x                                                                                              )                                                        +                                                            ∂                                              ∂                        y                                                              ⁢                                          (                                                                        I                          ⁡                                                      (                                                          x                              ,                                                                                                                          ⁢                              y                                                        )                                                                          ⁢                                                                              ∂                                                          ϕ                              ⁡                                                              (                                                                  x                                  ,                                                                                                                                          ⁢                                  y                                                                )                                                                                                                                          ∂                            y                                                                                              )                                                                      =                                                                                                    I                        ⁡                                                  (                                                      x                            ,                                                                                                                  ⁢                            y                                                    )                                                                    ⁢                                              (                                                                                                                                            ∂                                2                                                            ⁢                                                              ϕ                                ⁡                                                                  (                                                                      x                                    ,                                                                                                                                                  ⁢                                    y                                                                    )                                                                                                                                                    ∂                                                              x                                2                                                                                                              +                                                                                                                    ∂                                2                                                            ⁢                                                              ϕ                                ⁡                                                                  (                                                                      x                                    ,                                                                                                                                                  ⁢                                    y                                                                    )                                                                                                                                                    ∂                                                              y                                2                                                                                                                                    )                                                              +                                                                                            ∂                                                      I                            ⁡                                                          (                                                              x                                ,                                                                                                                                  ⁢                                y                                                            )                                                                                                                                ∂                          x                                                                    ⁢                                                                        ∂                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                                                                                                                  ⁢                                y                                                            )                                                                                                                                ∂                          x                                                                                      +                                                                                            ∂                                                      I                            ⁡                                                          (                                                              x                                ,                                                                                                                                  ⁢                                y                                                            )                                                                                                                                ∂                          y                                                                    ⁢                                                                        ∂                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                                                                                                                  ⁢                                y                                                            )                                                                                                                                ∂                          y                                                                                                      =                                                                                    I                        ⁡                                                  (                                                      x                            ,                                                                                                                  ⁢                            y                                                    )                                                                    ⁢                                                                        ∇                          2                                                ⁢                                                  ϕ                          ⁡                                                      (                                                          x                              ,                                                                                                                          ⁢                              y                                                        )                                                                                                                +                                                                  ∇                                                  I                          ⁡                                                      (                                                          x                              ,                                                                                                                          ⁢                              y                                                        )                                                                                              ·                                              ∇                                                  ϕ                          ⁡                                                      (                                                          x                              ,                                                                                                                          ⁢                              y                                                        )                                                                                                                                                                                                      (        3        )            Where
      ∇    2    ⁢      =                            ∂          2                          ∂                      x            2                              +                        ∂          2                          ∂                      y            2                              
Approximating the second term on the right side of expression (3) to zero, the approximation expression expressed by the following expression (4) is obtained.
                                          ∂                          I              ⁡                              (                                  x                  ,                                                                          ⁢                  y                                )                                                          ∂            z                          ≅                              -                                                            I                  ⁡                                      (                                          x                      ,                                                                                          ⁢                      y                                        )                                                  ⁢                                                                              κ                                ⁢                                    ∇              2                        ⁢                          ϕ              ⁡                              (                                  x                  ,                                                                          ⁢                  y                                )                                                                        (        4        )            In expression (4), φ(x,y) can be obtained from I(x,y) by a solution method such as the finite element method.
However, in the approximation expression (4), there is a problem that the estimated accuracy of the phase φ(x,y) becomes low when the approximation on the second term ∇I(x,y)·∇φ(x,y) included in expression (3) to zero is not appropriate.
Further, using a differential coefficient of intensity obtained from two pieces of detection data instead of the left side of expression (4), a phase can be restored by using at least two images. However, there is a problem that, when the phase is restored from a small number of images as many as two, the estimated accuracy of the phase becomes low in the case where the images are degraded due to noise etc.